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From: Alan Mackenzie <acm@muc.de>
Newsgroups: sci.math
Subject: Re: The non-existence of "dark numbers"
Date: Fri, 14 Mar 2025 13:35:27 -0000 (UTC)
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WM <wolfgang.mueckenheim@tha.de> wrote:
> On 13.03.2025 18:53, Alan Mackenzie wrote:
>> WM <wolfgang.mueckenheim@tha.de> wrote:

>> "Definable number" has not been defined by you, except in a sociologic=
al
>> sense.

> Then use numbers defined by induction:

> |=E2=84=95 \ {1}| =3D =E2=84=B5o.
> If |=E2=84=95 \ {1, 2, 3, ..., n}| =3D =E2=84=B5o
> then |=E2=84=95 \ {1, 2, 3, ..., n+1}| =3D =E2=84=B5o.

> Here the numbers n belonging to a potentially infinite set are defined.=
=20
> This set is called =E2=84=95_def.

You're confusing yourself with the outdated notion "potentially
infinite".  The numbers n in an (?the) inductive set are N, not N_def.
Why do you denote the natural numbers by "N_def" when everybody else just
calls them "N"?

> It strives for =E2=84=95 but never reaches it because .....

It doesn't "strive" for N.  You appear to be thinking about a process
taking place in time, whereby elements are "created" one per second, or
whatever.  That is a wrong and misleading way of thinking about it.  The
elements of N are defined and proven to exist.  There is no process
involved in this.

>>> =E2=88=80n =E2=88=88 =E2=84=95_def: |=E2=84=95 \ {1, 2, 3, ..., n}| =3D=
 =E2=84=B5o infinitely many
>>> numbers remain. That is the difference between dark and definable
>>> numbers.

>> Rubbish!  It's just that the set difference between an infinite set an=
d a
>> one of its finite subsets remains infinite.

> Yes, just that is the dark part. All definable numbers belong to finite=
=20
> sets.

Gibberish.  What does it mean for a number to "belong to" a finite set?
If you just mean "is an element of", then it's trivially true, since any
number n is a member of the singleton set {n}.

>> That doesn't shed any light on "dark" or "defi[n]able" numbers.

> Du siehst den Wald vor B=C3=A4umen nicht.
> [ You can't see the wood for the trees. ]

>>> =E2=84=95_def is a subset of =E2=84=95. If =E2=84=95_def had a last
>>> element, the successor would be the first dark number.

>> If, if, if, ....  "N_def" remains undefined, so it is not sensible to
>> make assertions about it.

> See above. Every inductive set (Zermelo, Peano, v. Neumann) is definabl=
e.

"Definable" remains undefined, so there's no point to answer here.  Did
Zermelo, Peano, or von Neumann use "definable" the way you're trying to
use it, at all?

>>>> But I can agree with you that there is no first "dark number".  That
>>>> is what I have proven.  There is a theorem that every non-empty
>>>> subset of the natural numbers has a least member.

>>> That theorem is wrong in case of dark numbers.

>> That's a very bold claim.  Without further evidence, I think it's fair
>> to say you are simply mistaken here.

> The potentially infinite inductive set has no last element. Therefore=20
> its complement has no first element.

You're letting "potentially infinite" confuse you again.  The inductive
set indeed has no last element.  So "its complement" (undefined unless we
assume a base set to take the complement in), if somehow defined, is
empty.  The empty set has no first element.

>>>>> When |=E2=84=95 \ {1, 2, 3, ..., n}| =3D =E2=84=B5o, then |=E2=84=95=
 \ {1, 2, 3, ..., n+1}| =3D
>>>>> =E2=84=B5o. How do the =E2=84=B5o dark numbers get visible?

>> There are no such things as "dark numbers", so talking about their
>> visibility is not sensible.

> But there are =E2=84=B5o numbers following upon all numbers of =E2=84=95=
_def.

N_def remains undefined, so talk about numbers following it is not
sensible.

>>>> There is no such thing as a "dark number".  It's a figment of your
>>>> imagination and faulty intuition.

>>> Above we have an inductive definition of all elements which have
>>> infinitely many dark successors.

>> "Dark number" remains undefined, except in a sociological sense.  "Dar=
k
>> successor" is likewise undefined.

> "Es ist sogar erlaubt, sich die neugeschaffene Zahl =CF=89 als Grenze z=
u=20
> denken, welcher die Zahlen =CE=BD zustreben, wenn darunter nichts ander=
es=20
> verstanden wird, als da=C3=9F =CF=89 die erste ganze Zahl sein soll, we=
lche auf=20
> alle Zahlen =CE=BD folgt, d. h. gr=C3=B6=C3=9Fer zu nennen ist als jede=
 der Zahlen =CE=BD."=20
> E. Zermelo (ed.): "Georg Cantor =E2=80=93 Gesammelte Abhandlungen mathe=
matischen=20
> und philosophischen Inhalts", Springer, Berlin (1932) p. 195.
> [ "It is even permissible to think of the newly created number as a
> limit to which the numbers nu tend.  If nothing else is understood,
> it's held to be the first integer which follows all numbers nu, that
> is, is bigger than each of the numbers nu." ]

> Between the striving numbers =CE=BD and =CF=89 lie the dark numbers.

That contradicts the long excerpt from Cantor you've just cited.
According to that, omega is the _first_ number which follows the numbers
nu.  I.e., there is nothing between nu (which we can identify with N) and
omega.  There is no place for "dark numbers".

>>> The set =E2=84=95_def defined by induction does not include =E2=84=B5=
o undefined numbers.

>> The set N doesn't include ANY undefined numbers.

>  =E2=84=B5o

>>>> Quite aside from the fact that there is no mathematical definition
>>>> of a "defined" number.  The "definition" you gave a few posts back
>>>> was sociological (talking about how people interacted with
>>>> eachother) not mathematical.

>>> Mathematics is social, even when talking to oneself. Things which can=
not
>>> be represented in any mind cannot be treated.

>> Natural numbers can be "represented in a mind", in fact in any
>> mathematician's mind.

> Not those which make the set =E2=84=95 empty by subtracting them
> =E2=88=80n =E2=88=88 =E2=84=95_def: |=E2=84=95 \ {1, 2, 3, ..., n}| =3D=
 =E2=84=B5o

That nonsense has no bearing on the representability of natural numbers
in a mathematician's mind.  You're just saying that the complement in N
of a finite subset of N is of infinite size.  Yes, and.... ?

> like the dark numbers can do
> =E2=84=95 \ {1, 2, 3, ...} =3D { }.

Dark numbers remain undefined.  The above identity, more succinctly
written as N \ N =3D { } holds trivially, and has nothing to say about th=
e
mythical "dark numbers".

> Regards, WM

--=20
Alan Mackenzie (Nuremberg, Germany).